Tuning, Temperament and the Harmonic Series

Tuning of musical scales is a matter that has (over a long period of
development) resulted in a great many systems, among them the western
twelve-tone, equal tempered scales. Temperament can only be understood in
relation to the pure intervals that are attributed to ancient greek
mathematician Pythagoras and his concept of »Harmony of the
Spheres«.

Aside from tuning issues the perceived sound of instruments is related to
the harmonic content of their sound waves. Wavetable and some additive
synthesizers have a very distinct sound and I've come to the conclusion that
they owe this to the perfect harmonic series they produce. The fixed phase
relation might also have some influence, but this would only be noticeable
when the the spectral amplitudes are modulated.

Spectral Cues of various Wavetables

Starting with the fundamental at the bottom and the first wave in the table
at the left, each pixel in the images represents the spectral amplitude of a
single partial. The color of the pixel encodes the sign (blue is negative and
red is positive), while the intensity encodes the magnitude. The magnitude
information has been compressed with a function (asinh) that behaves linearly
about zero and logarithmic for larger values, because otherwise you wouldn't
be able to see much detail. The greyscale images drop the sign
information. Depending on screen resolution the images are best viewed at 3x
to 4x magnification. Make sure you viewer does not smooth scaled pictures, you
should see »blocky« images.

with sign magnitude information or
only with magnitude
informationIf you look closely, you can see several synthetic
spectra from »hand-drawn« waveforms. Please note that the
waves do not necessarily appear in the order they would be used in a
wavetable. The SQ80 of course does not have »real«
wavetables, but most waves are very short — 256 samples for a
single-cycle waveform and 4096 samples for, well, sample waves (which
have been split into 16 consecutive waves for the wavetable
conversion) — and therefore it sounds very similar…

Continously variable saw wave and a spectrally
enhanced version

Continously variable saw
wave with sign magnitude
information or only with
magnitude informationSpectrally
enhanced continously variable saw wave with sign magnitude information or
only with magnitude
informationEverybody knows how nice pulsewidth modulation can
sound. What if we would morph the waveform from downward saw to symmetric
triangle and further to upward saw and back? The answer is that we get a
similarly pleasing effect. However the symmetric triangle wave, being a close
relative to the sine wave has a much faster harmonic decay than the saw
wave. This makes the endpoints of the modulation stand out sonically even if
the sum of all spectral energies is scaled to be constant (this results in
different amplitudes of the waves, but keeps a certain notion of
»loudness« constant). We can fix that by making the slopes of the
wave more and more concavely curved as we get away from the saw waves at the
end of the modulation. That transfers more of the spectral energy to the
higher harmonics, so that the character of the sound does not change so
drastically over the full modulation width and is percieved with almost
constant loudness.

Scales and Harmonics

The relation of the various scales and harmonic series can perhaps be more
easily understood from this visualization [PDF 9k][PNG
11k] showing the harmonic relation on a musical scale; it's still a very
busy chart. As guidelines the equal tempered scale and the scale resulting
from the pythagorean pure intervals are shown. A pure or pythagorean interval
is one where the frequency ratio is of two small integers. The pythagorean
scale
c : d : e : f : g : a : b : c
is given by
1/1 : 9/8 : 5/4 : 4/3 : 3/2 : 5/3 : 15/8 : 2/1. The
most pure interval aside from the octave is hence the pythagorean fifth, given
by a frequency ratio of 3/2. A comparison of progressions of pure intervals
reveals that it is (mathematically) impossible to arrive at exactly the same
point: A circle of twelve pythagorean fifths is about 74/73 sharp of seven
pythagorean octaves, the difference is known as the pythagorean
comma. Likewise four pythagorean fifths are 81/80 sharp of a pythagorean third
two octaves up, known as the syntonic comma. Looking at the chart you'll find
similar »commas« with the other pythagorean intervals, most
notably the fourth (I don't know if these »commas« have their own
names).

The second part of the chart shows the series of 255 perfect harmonics and
a series of 133 »stretched« harmonic, relating to an ideal string
(and wavetable synthesis) and a stiff, but massless string. The amount of
stretch (or dispersion) in the latter has been arbitrarily set to amount to
about 33 cent (a third of a semitone) over four octaves, a number that seems
to be agreed upon for upright piano strings. The higher harmonics are almost
certainly wrong for a piano, as a piano string has finite mass. Mass leads to
a compression of harmonics, but does not dominate over the stretching effect
of the stiffness at least in the audio range. I haven't got hold of a spectrum
for a piano string that would allow to assess the relative importance of mass
and dispersion - please let me know if you have one. The massless
approximation seems to be good enough for a Violin string, see the section on
»Modeling the stiffness of the string« in the paper Impact of
String Stiffness on Virtual Bowed Strings by Stefania Serafin and J. O. Smith.